Testing hypotheses via a mixture estimation model

TL;DR

This paper introduces a Bayesian hypothesis testing approach using mixture models, enabling model comparison through estimation of mixture weights, which addresses issues like improper priors and the Lindley-Jeffreys paradox.

Contribution

It proposes a novel mixture estimation framework for Bayesian testing that allows improper priors and resolves classical paradoxes, with analysis of prior sensitivity and convergence properties.

Findings

Posterior weights become insensitive to prior choice as sample size grows.

Using a Beta(0.5, 0.5) prior is recommended due to its robustness.

The convergence rates of posterior weights and probabilities are similar.

Abstract

We consider a novel paradigm for Bayesian testing of hypotheses and Bayesian model comparison. Our alternative to the traditional construction of posterior probabilities that a given hypothesis is true or that the data originates from a specific model is to consider the models under comparison as components of a mixture model. We therefore replace the original testing problem with an estimation one that focus on the probability weight of a given model within a mixture model. We analyze the sensitivity on the resulting posterior distribution on the weights of various prior modeling on the weights. We stress that a major appeal in using this novel perspective is that generic improper priors are acceptable, while not putting convergence in jeopardy. Among other features, this allows for a resolution of the Lindley-Jeffreys paradox. When using a reference Beta B(a,a) prior on the mixture weights, we note that the sensitivity of the posterior estimations of the weights to the choice of a vanishes with the sample size increasing and avocate the default choice a=0.5, derived from Rousseau and Mengersen (2011). Another feature of this easily implemented alternative to the classical Bayesian solution is that the speeds of convergence of the posterior mean of the weight and of the corresponding posterior probability are quite similar.